From Theorem REF  and by [1]}, a strongly continuous evolution family \(\lbrace E_0(t,s)\rbrace \subset \mathcal {L}(V)\)  exists such that \(u(t):=E_0(t,s)x\)  is the unique solution in \(H^1 ([0,T],H) \cap L^2([0,T],V)\)  to the Cauchy problem (REF ).
Since \(A(t)\)  is accretive, it follows that
\(\Vert  u(t) \Vert _H^2 - \Vert u(s)\Vert _H^2 && = \int _s^t \frac{d}{dt} \Vert  u(\tau ) \Vert _H^2 d\tau = 2 \int _s^t \Re ( \langle u^{\prime }(\tau ) , u(\tau ) \rangle ) d\tau \\\ && = 2 \int _s^t - \Re ( \langle A(\tau ) u(\tau ), u(\tau ) \rangle ) d\tau \le 0,\) 
which readily implies that, for any fixed \((t,s)\in \Delta \) , \(\Vert u(t)\Vert _H=\Vert  E_0(t,s) x \Vert _H \le \Vert  x \Vert _H=\Vert u(s)\Vert _H.\)  This last inequality, together with the density of \(V\)  in \(H\) , permit to uniquely extend \(E_0(t,s): V \rightarrow H\)  to a linear operator \(E(t,s):H\rightarrow H\)  by means of the BLT Theorem. The strong continuity of the family \(\lbrace  E(t,s) \rbrace \)  also follows from a density argument.
(i). The claims follow from above construction and (REF ) of Remark REF .
(ii).
Arguing as in [1]}, fix \(x\in V\)  and set \(v(t):= t E(t,0)x.\)  Then, since \(E(\cdot , 0)x\in L^2([0,T],H)\)  and by Theorem REF , \(v\)  is the unique solution of the nonhomogeneous Cauchy problem
\({\left\lbrace \begin{array}{ll}v^{\prime }(t)+{A}(t)v(t)=E (t,0)x, \qquad a.e. \ \ t\in [0,T]\ \\v(0)=0,\end{array}\right.}\) 
moreover by (REF ),
\(\Vert v\Vert _{H^{1}([0,T],H)}+\Vert v\Vert _{L^2([0,T],V)}\\+\Vert A(\cdot )v(\cdot )\Vert _{L^2([0,T],H)} & \le & C\left(\Vert E(\cdot ,0)x\Vert _{L^2([0,T],H)}\right)\\& \le & CT^{1/2}\Vert x\Vert _{H}.\) 
This last fact means that for any \(\varepsilon \in (0,T),\)
\(\nonumber \Vert E(\cdot ,0)x\Vert _{L^2([\varepsilon ,T],H)}+\Vert E(\cdot ,0)x\Vert _{L^2([\varepsilon ,T],V)}\\\nonumber +\left\Vert \frac{dE (\cdot ,0)x}{dt}\right\Vert _{L^2([\varepsilon ,T],H)} & \le & \frac{CT^{1/2}}{\varepsilon }\Vert  x \Vert _H.\) 
Again, by exploiting the density of \(V\)  in \(H,\)  it follows that the same inequalities also holds for any fixed \(x\in H.\)  We have then proved that \(E(t,0)x\)  lies in \(H^1_\text{loc} ([0,T],H) \cap L^2_\text{loc}([0,T],V)\cap C([0,T],H)\)  as well as the estimate (REF ).
(iii). It essentially follows as [1]} for \(p=2.\)
(iv). By maximal regularity, \(u\)  has a continuous representation in \(C([0,T],H)\)  (see [4]}) and \(u\)  solves (REF ) in \(H^1_\text{loc} ([0,T],H) \cap L^2_\text{loc}([0,T],V)\cap C([0,T],H).\)  Then result derives from the uniqueness of the solution.
